Mathematical logic with diagrams dr frithjof dau

Mathematical Logic
with Diagrams
Based on the Existential Graphs of Peirce
Frithjof Dau, TU Dresden, Germany


2

Disclaimer: This is (nearly) the final version of this treatise. There will
be no more content added. It is only subject of a further proof-reading. For
this reason, if you find any misspellings, gaps, flaws, etc., please contact me
(). Similarly, do not hesitate to contact me if you have any
questions.
Frithjof Dau, January 23, 2008


Come on, my Reader, and let us construct a diagram to illustrate
the general course of thought; I mean a system of diagrammatization by means of which any course of thought can be represented
with exactitude.
Peirce, Prolegomena to an Apology For Pragmaticism, 1906



Contents

Start
1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1 The Purpose and the Structure of this Treatise . . . . . . . . . . . . . .

3

Short Introduction to Existential Graphs . . . . . . . . . . . . . . . . . .

7

2.1 Alpha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2 Beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3

Theory of Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Icons, Indices, Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Types and Tokens, Signs and Replicas . . . . . . . . . . . . . . . . . . . . . 23

4

The Role of Existential Graphs in Peirce’s Philosophy . . . . . 25
4.1 Foundations of Knowledge and Reasoning . . . . . . . . . . . . . . . . . . 26
4.2 Existential Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5

Formalizing Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1 Problems with Existential Graphs Replicas . . . . . . . . . . . . . . . . . 40
5.2 The First Approach to Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Linear Representations of Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 The Second Approach to Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 52


6

6

Contents

Some Remarks to the Books of Zeman, Roberts, and Shin . 55

Alpha
7

Syntax for Alpha Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8

Semantics and Calculus for Formal Alpha Graphs . . . . . . . . . 75
8.1 Semantics for Formal Alpha Graphs . . . . . . . . . . . . . . . . . . . . . . . 75
8.2 Some Remarks to the Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.3 Calculus for Alpha Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.4 Some Simple Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9

Soundness and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.1 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

10 Translation to Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . 89

Beta
11 Getting Closer to Syntax and Semantics of Beta . . . . . . . . . . . 95
11.1 Lines of Identities and Ligatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
11.2 Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
11.3 Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11.4 Border Cases: LoIs Touching a Cut or Crossing on a Cut . . . . . 116
12 Syntax for Existential Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
12.1 Relational Graphs with Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
12.2 Existential Graph Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
12.3 Further Notations for Existential Graph Instances . . . . . . . . . . . 132
12.4 Formal Existential Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
13 Semantics for Existential Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 139
13.1 Semantics for Existential Graph Instances . . . . . . . . . . . . . . . . . . 139
13.2 Semantics for Existential Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 143


Contents

7

14 Getting Closer to the Calculus for Beta . . . . . . . . . . . . . . . . . . . . 147

14.1 Erasure and Insertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
14.2 Iteration and Deiteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
14.3 Double Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
14.4 Inserting and Deleting a Heavy Dot . . . . . . . . . . . . . . . . . . . . . . . . 160
15 Calculus for Formal Existential Graphs . . . . . . . . . . . . . . . . . . . . 161
16 Improving the Handling of Ligatures . . . . . . . . . . . . . . . . . . . . . . 167
16.1 Derived Rules For Ligatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
16.2 Improving the Reading of Ligatures . . . . . . . . . . . . . . . . . . . . . . . . 176
17 Soundness of the Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
18 First Order Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
18.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
18.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
18.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
19 Syntactical Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
19.1 Definition of Φ and Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
19.2 Semantical Equivalence between Graphs and Formulas . . . . . . . 210
20 Syntactical Equivalence to F O and Completeness . . . . . . . . . . 215
20.1 Ψ Respects

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

20.2 Identity of G and Ψ (Φ(G)) and Completeness of

. . . . . . . . . . . 225

21 Working with Diagrams of Peirce’s Graphs . . . . . . . . . . . . . . . . 227

Extending the System
22 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
23 Adding Objects and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

23.1 General Logical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
23.2 Extending the Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247


23.3 Examples for EGIs with Objects and Functions . . . . . . . . . . . . . 251
24 Vertices with Object Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
24.1 Syntax and Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
24.2 Correspondence between vertex-based EGIs and EGIs . . . . . . . . 260
24.3 Calculus for vertex-based EGIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
24.4 Ligatures in vertex-based EGIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
25 Relation Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
25.1 Semi Relation Graph Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
25.2 Relation Graph Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
26 Peirce’s Reduction Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
26.2 Peircean Algebraic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
26.3 Graphs for Peircean Algebraic Logic . . . . . . . . . . . . . . . . . . . . . . . 297
26.4 Peirce’s Reduction Thesis for Relation Graphs . . . . . . . . . . . . . . 307
26.5 The Contributions of Herzberger and Burch . . . . . . . . . . . . . . . . 312

Appendix
List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327


Start

1
Introduction

The research field of diagrammatic reasoning investigates all forms of human
reasoning and argumentation wherever diagrams are involved. This research
area is constituted from multiple disciplines, including cognitive science and
psychology as well as computer science, artificial intelligence, mathematics
and logic. However, it should not be overlooked that there has been until
today a long-standing prejudice against non-symbolic representation in mathematics and logic. Without doubt, diagrams are often used in mathematical
reasoning, but usually only as illustrations or thought aids. Diagrams, many
mathematicians say, are not rigorous enough to be used in a proof, or may
even mislead us in a proof. This attitude is captured by the quotation below:
[The diagram] is only a heuristic to prompt certain trains of inference;
... it is dispensable as a proof-theoretic device; indeed ... it has no
proper place in a proof as such. For the proof is a syntactic object
consisting only of sentences arranged in a finite and inspectable area.
Neil Tennant 1991, quotation adopted from [Bar93]
Nonetheless, there exist some diagrammatic systems which were designed for
mathematical reasoning. Well-known examples are Euler circles and Venn diagrams. More important to us, at the dawn of modern logic, two diagrammatic
systems had been invented in order to formalize logic. The first system is
Frege’s Begriffsschrift, where Frege attempted to provide a formal universal
language. The other one is the systems of existential graphs (EGs) by Charles
Sanders Peirce, which he used to study and describe logical argumentation.
But none of these systems is used in contemporary mathematical logic. In
contrast: for more than a century, linear symbolic representation systems (i.e.
formal languages which are composed of signs which are a priori meaningless,
and which are therefore manipulated by means of purely formal rules) have
been the exclusive subject for formal logic. There are only a few logicians who

2

1 Introduction

have done research on formal, but non-symbolic logic. The most important
ones are without doubt Barwise and Etchemendy. They say that
there is no principle distinction between inference formalisms that use
text and those that use diagrams. One can have rigorous, logically
sound (and complete) formal systems based on diagrams.
Barwise and Etchemendy 1994, quotation adopted from [Shi02a]
This treatise advocates this view that rigor formal logic can be carried out
by means of manipulating diagrams. In order to do this, the systems of existential graphs is elaborated in a manner which suits the needs and rigor of
contemporary mathematics.
There are good reasons for choosing Peirce’s EGs for the purpose of this
treatise. Peirce had been a philosopher and mathematician who devoted his
life to the investigation of reasoning and the growth of knowledge. He was
particularly interested in the exploration of mathematical reasoning, and EGs
are designed as an instrument for the investigation of such reasoning.
Before he invented EGs at the end of the 19th century, Peirce contributed
much to the development of the symbolic approach to mathematical logic.
1
Thus, Peirce was very familiar with both approaches – the diagrammatic
and the symbolic – to logic. As he was interested in an instrument for the
investigation of logic (which has to be distinguished from the investigation
and development of logic as such), he discussed the differences, the advantages
and disadvantages, of these two approaches to a large extent. Particularly, he
elaborated a comprehensive theory of what he already called diagrammatic
reasoning, and he considered his diagrammatic system of EGs to be far more
perfect for the investigation of mathematical reasoning than the symbolic

approach he developed as well. His philosophical considerations, his arguments
for his preference of the diagrammatic approach to logic, will give us valuable
insights to how logic with diagrams can be done. This is a first reason to
particularly choose EGs for elaborating formal logic by means of diagrams.
The system of EGs is divided into three parts which are called Alpha, Beta
and Gamma. Beta builds upon Alpha, and Gamma in turn builds upon Beta.
As EGs are an instrument for the investigation of mathematical reasoning,
it is not surprising that the different parts of EGs correspond to specific
fragments of mathematical logic. It is well accepted that Alpha corresponds
to propositional logic, and Beta corresponds to first-order predicate logic.2
1

2

For example, he invented, independently from Frege, together with his student
O. H. Mitchell a notation for existential and universal quantification. According
to Putnam [Put82], Frege discovered the quantifier four years before Mitchell, but
’Frege did “discover” the quantifier in the sense of having the rightful claim to
priority; but Peirce and his students discovered it in the effective sense.’
Later, it will be discussed in more detail how far the arguments which are given
for this claim can be understood as strict, mathematical proofs.


1.1 The Purpose and the Structure of this Treatise

3

Gamma is more complicated: It contains features of higher order and modal
logic, the possibilty to express self-reference, and other features. Due to its
complexity, it was not completed by Peirce. The majority of works which deal

with Gamma deal only with the fragment of Gamma which corresponds to
modal logic.
The formal mathematical logic we use nowadays emerged at the beginning
of the 20th century. Russell’s and Whitehead’s landmark work Principia
Mathematica, probably the most influential book on modern logic, had been
published in the years 1910–1912. It is obvious that Peirce’s works can by no
means satisfy the needs and criteria of present mathematical logic. His contributions to symbolic logic found their place in the development of modern
formal logic, but his system of EGs received no attention during this process.
Thus, in order to prove mathematically that Alpha and Beta correspond to
propositional and first order predicate logic, respectively, the system of EGs
has first to be be reworked and reformulated as a precise theory of mathematical logic. Then the correspondence to the symbolic logic we use nowadays
can be mathematically formulated and proven.
Several authors like Zeman, Roberts, Sowa, Burch or Shin have explored the
system of EGs. Most of them work out a correspondence of Alpha and Beta
to propositional and first order predicate logic, but it will be discussed later in
detail how far their arguments can be considered to be mathematical proofs.
Moreover, these authors usually fail to implement EGs as a logic system on its
own without a need for translations to other formal, usually symbolic logics,
that is, they fail to provide a dedicated, extensional semantics for the graphs.
The attempt of this treatise is to amend this gap. EGs will be developed
as a formal, but diagrammatic, mathematical logic, including a well-defined
syntax, an extensional semantics, and a sound and complete calculus. Translations from and to symbolic logic are provided as additional elements to work
out the correspondence between diagrammatic and symbolic logic in a mathematical fashion. The methodology of developing a formal, diagrammatic logic
is carried out on EGs, but it can be transferred to the development of different
forms of diagrammatic logic as well.

1.1 The Purpose and the Structure of this Treatise
The overall purpose of this treatise has already been explicated: it is to develop a general framework and methodology for a diagrammatic approach to
mathematical logic. In Chpt. 3, a small part of Peirce’s extensively developed
semiotics, i.e. theory of signs, is presented. This part is helpful to elaborate

the specific differences between symbolic and diagrammatic representations of
logic. Moreover, it gives us a first hint on how diagrams can be mathematically
formalized. This will be more thoroughly discussed in Chpt. 5. In this chapter,


4

1 Introduction

the use of representations in mathematical logic is investigated, and two different, possible approaches for a formalization of diagrams are discussed and
compared. From the results of this discussion, we obtain the methodology for
the formalization of diagrams which is be used in this treatise.
In the frame of this general purpose, Peirce’s EGs serve as a case-study. However, understanding EGs as a ‘mere’ case-study is much too narrow. It has
already been argued why it is convenient not to implement an ‘arbitrary’ diagrammatic system, but to consider especially Peirce’s EGs. Although they
are not completely independent from each other, there are two main lines in
the elaboration of Peirce’s EGs.
First of all, this treatise aims to describe Peirce’s deeper understanding of
his systems of EGs (this is similar to Robert’s approach in [Rob73]. See also
Chpt. 6). Due to this aim, in Chpt. 4 it is discussed which role Peirce’s systems
of EGs in his whole philosophy has (this chapter relies on Peirce’s semiotics
which is described in Chpt. 3, but it is a separate chapter in the sense that
the remaining treatise makes little reference to it), and Peirce’s philosophical
intention in the design of the syntax and the transformation rules of EGs is
discussed. For Peirce’s Beta graphs, in Chpt. 11, Peirce’s deeper understanding
on the form and meaning of his graphs is investigated, and in Chpt. 14, the
same is done for Peirce’s transformation rules. These four chapters offer a soto-speak ‘historical reconstruction’ of Peirce’s graphs. Chpts. 11 and 14 are
also needed for the second goal of this treatise: to rework Peirce’s graphs as a
system which fulfills the standards of our contemporary mathematical logic.
This is done first for Peirce’s Alpha graphs, then for his Beta graphs.
Alpha graphs are mathematically elaborated in Chpts. 8–10. The syntax of

these graphs is presented in Chpt. 7, the semantics and calculus is presented
in Chpt. 8. In Chpt. 9, it is directly shown that the calculus is sound and complete. Propositional logic is encompassed by first order logic; analogously, the
system of Alpha graphs is encompassed by the system of Beta graphs. Thus,
from a mathematical point of view, the separate elaboration of Alpha graphs
is not needed. Nonetheless, propositional logic and first order logic are the
most fundamental kinds of mathematical logic, thus, in most introductions to
mathematical logic, both kinds are separately described. Moreover, this treatise aims to formalize EGs, and Peirce separated EGs into the systems Alpha,
Beta and Gamma as well. For this reason, Alpha graphs are separately treated
in this treatise, too. So, Alpha of this treatise can be seen as a preparation
to Beta. Due to this reason, the formalization of Alpha graphs is geared to
the formalization of Beta graphs. In fact, the formalization of Alpha graphs is
somewhat a little bit too clumsy and technical. If one aims to develop solely
the Alpha graphs in a mathematical manner, their formalization could be
simplified, but in the light of understanding Alpha as a preparation for Beta,
the herein presented formalization is more convenient. Finally, in Chpt. 10,
translations between Alpha graphs and formulas of propositional logic are provided. It will be shown that these translations are meaning-preserving, thus


1.1 The Purpose and the Structure of this Treatise

5

we have indeed a correspondence between the system of Alpha graphs and
propositional logic.
Alpha graphs, that is their diagrammatic representations as well as the transformation rules, are somewhat easy to understand and hard to misinterpret.
First order logic is much more complex than propositional logic, henceforth,
Beta of this treatise is much more extensive than Alpha. Obtaining a precise
understanding of the diagrams of Beta graphs, as well as a precise understanding of the transformation rules, turns out to be much harder. This is
partly due to the fact that Alpha graphs are discrete structures, whereas Beta
graphs (more precisely: the networks of heavily drawn lines in Beta graphs)

are a priori non-discrete structures. For this reason, in Chpt. 11, the diagrams of Peirce’s Beta graphs are first investigated to a large degree, before
their syntax and semantics are formalized in Chpts. 12 and 13. It turns out
that EGs should be formalized as classes of discrete structures. Then, the
transformation rules for Peirce’s Beta graphs are first discussed separately in
Chpt. 14, before their formalization is provided in Chpt. 15. The soundness
of these rules can be shown similar to the soundness of their counterparts in
Alpha. This is done in Chpt. 17. Similar to Alpha, translations between Beta
graphs and formulas, now first order logic (FO), are provided. In Chpt. 18,
the style of FO which is used for this purpose is presented. In Chpt. 19, the
translations between the system of Beta graphs and FO are provided, and it
is shown that these translations are meaning-preserving. It remains to show
that the calculus for Beta graphs is complete (the completeness cannot be obtained from the facts that the translations are meaning-preserving). Proving
the completeness of logic system with the expressiveness of first order logic is
somewhat extensive. For this reason, in contrast to Alpha, the completeness
of Beta will not be shown directly. Instead, the well-known completeness of
a calculus for symbolic first order logic will be transferred to Beta graphs.
In Chpt. 20, it will be shown that the translation from formulas to graphs
respects the derivability relation as well, from which the completeness of the
calculus for Beta graphs is concluded. Finally in Chpt. 21, the results of the
preceeding chapters are transferred to the diagrammatic representations of
EGs. Thus, this chapter concludes the program of formalizing Peirce’s EGs.
In Chpts. 22–26, some extensions of EGs which slightly extend their expressiveness are investigated. First, an overview of them is provided in Chpt. 22.
In Chpts. 23 and 24, the graphs are augmented with objects and functions. In
Chpts. 25 and 26, a new syntactical device which corresponds to free variables
are added to the graphs. The resulting graphs evaluate to relations instead of
propositions and are therefore termed relation graphs. In Chpts. 23–25, the
syntax, semantics and calculus of Beta are appropriately extended to cover
the new elements. Instead of the logic of the extended graphs, Chpt. 26 focuses on operations on relations and on how these operations are reflected
by relation graphs. Then a mathematical version of Peirce’s famous reduction
thesis is proven for relation graphs.

6

1 Introduction

The aim and the structure of this treatise should be clear now. In the remainder of this section, some unusual features of treatise are explained.
First of all, this treatise contains a few definitions, lemmata and theorems
which cannot be considered to be mathematical. For example, this concerns
discussions of the relationship between mathematical structures and their representations. A ‘definition’ how a mathematical structure is represented fixes
a relation between these mathematical structures and their representations,
but as the representations are non-mathematical entities, this definition is not
a definition in a rigid mathematical sense. To distinguish strict mathematical
definitions for mathematical entities and definitions where non-mathematical
entities are involved, the latter will be termed Informal Definition. Examples
can be found in Def. 5.1 or Def. 7.8.
Second, there are some parts of the text that provide further discussions or
expositions which are not needed for the understanding of the text, but which
may be of interest for some readers. These parts can be considered to be
‘big footnotes’, but, due to their size, they are not provided as footnotes, but
embedded into the continuous text. To indicate them clearly, they start with
the word ‘Comment’ and are printed in footnote size. An example can be
found below.
Third, the main source of Peirce’s writings are the collected papers [HB35].
The collected papers are -as the name says- a thematically sorted collection
of his writings. They consist of eight books, and in each book, the paragraphs
are indexed by three-digit numbers. This index s adopted without explicitely
mentioning the collected papers. For example, a passage in this treatise like
‘in 4.476, Peirce writes [. . .]’ refers to [HB35], book 4, paragraph 476.
Comment: Unfortunately, the collected papers are by no means a complete collection

of Peirce’s manuscripts: more than 100.000 pages, archived in the Houghton Library
at Harvard, remain unpublished. Moreover, due to the attempt of the editors to
provide the writings in a thematically sorted manner, they divided his manuscripts,
placed some parts of them in different places of the collected papers, while other parts
are dismissed. Moreover, they failed to indicate which part of the collected papers is
obtained from which source, and sometimes it is even impossible to realize whether
a chapter or section in the collected papers is obtained from exactly one source
or it is assembled from different sources. As Mary Keeler writes in [Kee95]: The
misnamed Collected Papers [. . .] contains about 150 selections from his unpublished
manuscripts, and only one-fifth of them are complete: parts of some manuscripts
appear in up to three volumes and at least one series of papers has been scattered
throughout seven.

Finally, in a few places we refer to some writings of Peirce, as they are catalogued by Robin in [Rob67]. As usual, such an reference is preceeded by ‘MS’,
e.g., ‘MS 507’ refers to nine pages of existential graphs, as classified by Robin
in [Rob67].


2
Short Introduction to Existential Graphs

Modern formal logic is presented in a symbolic and linear fashion. That is,
the signs which are used in formal logic are symbols, i.e. signs which are a
priori meaningless and gain their meaning by conventions or interpretations
(in Chpt. 3, the term ‘symbol’ is discussed in detail). The logical propositions,
usually called formulas or sentences, are composed of symbols by writing them
-like text- linearly side by side (in contrast to a spatial arrangement of signs in
diagrams). In fact, nowadays formal logic seems to dismiss any non-symbolic
approach (see the discussion at the beginning of Chpt. 5), thus formal logic
is identified with symbolic logic.1

In contrast to the situation we have nowadays, the formal logic of the nineteenth century was not limited to symbolic logic only. At the end of that
century, two important diagrammatic systems for mathematical logic have
been developed. One of them is Frege’s Begriffsschrift. The ideas behind the
Begriffsschrift had an influence on mathematics which can hardly be underestimated, but the system itself had never been used in practice.2 The other
diagrammatic system are Peirce’s existential graphs, which are the topic of
this treatise. Before Peirce developed his diagrammatic form of logic, he contributed to the development of symbolic logic to a large extent. He contributed
to the algebraic notation for predicate logic to a large extent (see [Rob73] for a
historical survey of Peirce’s contributions to logic). For example, he invented
a notation for quantification. Although Peirce investigated the algebraic notation, he was not satisfied with this form of logic. As Roberts says in [Rob73]: It
is true that Peirce considered algebraic formulas to be diagrams of a sort; but
it is also true that these formulas, unlike other diagrams, are not ‘iconic’ —
that is, they do not resemble the objects or relationships they represent. Peirce
took this for a defect. Unfortunately, Peirce discovered his system of EGs at
1
2

A much more comprehensive discussion of this topic can be found in [Shi02a].
The common explanation for this is that Frege’s diagrams had been to complicated to be printed.


8

2 Short Introduction to Existential Graphs

the very end of the nineteenth century (in a manuscript of 1906, he says that
he invented this system in 1896. See [Rob73]), when symbolic logic already
had taken the vast precedence in formal logic. For this reason, although Peirce
was convinced that EGs are a much better approach to formal logic than any
symbolic notation of logic, EGs did not succeed against symbolic logic. It is
somewhat ironic that EGs have been ruled out by symbolic formal logic, a

kind of logic which was developed on the basis of Peirce’s algebraic notation
he introduced about 10 years before.
This treatise attempts to show that rigor formal logic can be carried out with
the non-symbolic EGs. Before we start with the mathematical elaboration of
EGs, in this chapter a first, informal introduction to EGs is provided.
The system of EGs is a highly elegant system of logic which covers propositional logic, first order logic and even some aspects of higher-order logic and
modal logic. It is divided into three parts: Alpha, Beta and Gamma.3 These
parts presuppose and are built upon each other, i.e. Beta builds upon Alpha,
and Gamma builds upon Alpha and Beta. In this chapter, Alpha and Beta
are introduced, but we will only take a short glance at Gamma.

2.1 Alpha
We start with the description of Alpha. The EGs of Alpha consist only of
two different syntactical entities: (atomar) propositions, and so-called cuts
(Peirce often used the term ‘sep’4 instead of ‘cut’, too) which are represented
by fine-drawn, closed, doublepoint-free curves.5 Atomar propositions can be
considered as predicate names of arity 0. Peirce called them medads.
Medads can be written down on an area (the term Peirce uses instead of ‘writing‘ is ‘scribing’ ). The area where the proposition is scribed on is what Peirce
called the sheet of assertion. It may be a sheet of paper, a blackboard, a
computer screen or any other surface. Writing down a proposition is to assert
it (an asserted proposition is called judgement). Thus,

it rains
is an EG with the meaning ‘it rains’, i.e. it asserts that it rains.
3

4

5

In [Pie04], Pietarinen writes that Peirce mentions in MS 500: 2-3, 1911, that he
even projected a fourth part Delta. However, Pietarinen writes that he found no
further reference to it. And, to the best of my knowledge, no other authors besides
Pietarinen have mentioned or even discussed Delta so far.
According to Zeman [Zem64], the term ‘sep’ is inspired from the latin term saepes,
which means ‘fence’. Before Fig. 2.3, a passage from Peirce is provided where he
writes that some cut is used to fence off a proposition from the sheet of assertion.
Double-point free means that the line must not cross itself.


2.1 Alpha

9

We can scribe several propositions onto the sheet of assertion, usually side by
side (this operation is called a juxtaposition). This operation asserts the
truth of each proposition, i.e. the juxtaposition corresponds to the conjunction of the juxtaposed propositions. For example, scribing the propositions ‘it
rains’, ‘it is stormy’ and ‘it is cold‘ side by side yields the graph
it rains

it is stormy

it is cold

which means ‘it rains, it is stormy and it is cold’. The propositions do not
have to be scribed or read off from left to right, thus
it is cold
it is stormy

it rains

is another possibility to arrange the same propositions onto the sheet of assertion, and this graph still has the meaning ‘it rains, it is stormy and it is
cold’.
Encircling a graph by a cut is to negate it. For example, the graph
it rains

has the meaning ‘it is not true that it rains’, i.e. ‘it does not rain’. The graph
it rains

it is cold

has the meaning ‘it is not true that it both rains and is cold’, i.e. ‘it does not
rain or it is not cold’ (the part ‘it is not true’ of this statement refers to the
whole remainder of the statement, that is, the whole proposition ‘it rains and
it is cold’ is denied.)
The space within a cut is called its close or area. Cuts may not overlap,
intersect, or touch6 , but they may be nested. The next graph has two nested
cuts.
it rains

it is stormy

it is cold

Its reading starts on the sheet of assertion, then proceeding inwardly. This way
of reading is called endoporeutic method by Peirce. Due to endoporeutic
6

This is not fully correct: Peirce often drew scrolls with one point of intersection
as follows:

. But in 4.474 he informs us that the node [the point of intersection] is of no particular significance, and a scroll may equally well be drawn
as

.


10

2 Short Introduction to Existential Graphs

reading, this graph has the meaning ‘it is not true that it rains and it is stormy
and that it is not cold’, i.e. ‘if it rains and if its stormy, then it is cold’. It has
three distinct areas: the area of the sheet of assertion contains the outer cut,
the area of the outer cut contains the propositions ‘it rains’ and ‘it is stormy’
and the inner cut, and the inner cut contains the proposition ‘it is cold’. An
area is oddly enclosed if it is enclosed by an odd number of cuts, and it is
evenly enclosed if it is enclosed by an even number of cuts. Thus, the sheet
of assertion is evenly enclosed, the area of the outer cut is oddly enclosed, and
the area of the inner cut is evenly enclosed. Moreover, for the items on the
area of a cut (or the area of the sheet of assertion), we will say that these
items are directly enclosed by the cut. Items which are deeper nested are
said to be indirectly enclosed by the cut. For example, the proposition
‘it is cold’ is directly enclosed by the inner cut and indirectly enclosed by the
outer cut.
The device of two nested cuts is called a scroll. From the last example we
learn that a scroll can be read as an implication. A scroll with nothing on
its outer area is called double cut. Obviously, it corresponds to a double
negation.
As we have the possibility to express conjunction and negation of propositions,
we see that Alpha has the expressiveness of propositional logic. Peirce also

provided a calculus for EGs (due to philosophical reasons, Peirce would object
against the term ‘calculus’. This will be elaborated in Chpt. 4). This calculus
has a set of five rules, which are named erasure, insertion, iteration,
deiteration, and double cut, and only one axiom, namely the empty
sheet of assertion. Each rule acts on a single graph. For Alpha, these rules can
be formulated as follows:
• Erasure: any evenly enclosed subgraph7 may be erased.
• Insertion: any graph may be scribed on any oddly enclosed area.
• Iteration: if a subgraph G occurs on the sheet of assertion or in a cut, then
a copy of the graph may be scribed on the same or any nested area which
does not belong to G.
• Deiteration: any subgraph whose occurrence could be the result of iteration
may be erased.
• Double Cut: any double cut may be inserted around or removed from any
area.
We will prove in this treatise that this set of rules is sound and complete. In
the following, a simple example of a proof (which is an instantiation of modus
ponens in EGs) is provided. Let us start with the following graph:
7

The technical term ‘subgraph’ will be precisely elaborated in Chpt. 7.


2.2 Beta

it rains

it is stormy

it rains

it is stormy

11

it is cold

It has the meaning ‘it both rains and is stormy, and if it both rains and is
stormy, then it is cold’. Now we see that the inner subgraph it rains it is stormy
may be considered to be a copy of the outer subgraph it rains it is stormy .
Hence we can erase the inner subgraph using the deiteration-rule. This yields:

it rains

it is stormy

it is cold

This graph contains a double cut, which now may be removed. We get:
it rains

it is stormy

it is cold

Finally we erase the subgraph it rains it is stormy with the erasure-rule and
get:
it is cold
So the graph with the meaning ‘it rains and it is stormy, and if it rains and
it is stormy, then it is cold’ implies the graph with the meaning ‘it is cold’.

2.2 Beta
The step from Alpha to Beta corresponds to the step from propositional logic
to first order logic. First of all, a new symbol, the line of identity, is
introduced. Lines of identity are used to denote both the existence of objects
and the identity between objects. They are represented as heavily drawn lines.
Second, instead of only considering medads, i.e. predicate names of arity 0,
now predicate names of arbitrary arity may be used.
Consider the following graph:

cat

on

mat

It contains two lines of identity, hence it denotes two (not necessarily different)
objects. The first line of identity is attached to the unary predicate ‘cat’, hence
the first object denotes a cat. Analogously the second line of identity denotes a
mat. Both lines are attached to the dyadic predicate ‘on’, i.e. the first object
(the cat) stands in the relation ‘on’ to the second object (the mat). The
meaning of the graph is therefore ‘there are a cat and a mat such that the cat
is on the mat’, or in short: a cat is on a mat. Analogously,


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2 Short Introduction to Existential Graphs

cat

table

between

door

means ‘there is a cat between a table and a door’.
Lines of identity may be composed to networks. Such a network of lines of
identity is called ligature. For example, in

cat
young
cute
we have a ligature composed of three lines of identity, which meet in a socalled branching point. Still this ligature denotes one object: the meaning
of the graph is ‘there is an object which is a cat, young and cute’, or ‘ there
is a cute and young cat’ for short.
Ligatures may cross cuts (it will become clear in Chpt. 11 why the term
‘ligature’ is used in these examples instead of writing that that lines of identity
may cross cuts). Consider the following graphs:

cat

cat

cat

The meaning of the first graph is clear: it is ‘there is a cat’. The second graph
is built from the first graph by drawing a cut around it, i.e. the first graph is
denied. Hence the meaning of the second graph is ‘it is not true that there is
a cat’, i.e. ‘there is no cat’. In the third graph, the ligature starts on the sheet

of assertion. Hence the existence of the object is asserted and not denied. For
this reason the meaning of the third graph is ‘there is something which is not
a cat’.
A heavily drawn line which traverses a cut denotes the non-identity of the
extremities of that line (again this will be discussed in Chpt. 11). For example,
the graph

cat

cat

has the meaning ‘there is an object o1 which is a cat, there is an object o2
which is a cat, and o1 and o2 are not identical’, that is, there are at least two
cats.
Now we have the possibility to express existential quantification, predicates
of arbitrary arities, conjunction and negation. Hence we see that Beta corresponds to first order predicate logic (that is first order logic with identity and
predicate names, but without object names or function names).


2.2 Beta

13

Essentially, the rules for Beta are extensions of the five rules for Alpha such
that the Beta rules cover the properties of the lines of identity. The Beta rules
are as follows:
• Erasure: any evenly enclosed subgraph and any evenly enclosed portion of
a line of identity may be erased.
• Insertion: any graph may be scribed on any oddly enclosed area, and two
portions of two lines of identity which are oddly enclosed on the same area

may be joined.
• Iteration: For a subgraph G on the sheet of assertion or in a cut, a copy
of this subgraph may be scribed on the same or any nested area which
does not belong to G. In this operation, it is allowed to connect any line
of identity of G, which is not scribed on the area of any cut of G, with
its iterated copy. Consequently, it is allowed to add new branches to a
ligature, or to extend any line of identity inwards through cuts.
• Deiteration: any subgraph whose occurrence could be the result of an
iteration may be erased.
• Double Cut: any double cut may be inserted around or removed from any
area. This transformation is still allowed if we have ligatures which start
outside the outer cut and pass through the area of the outer cut to the are
of the inner cut.
The precise understanding of these rules will be unfolded in Chpt. 14. In this
chapter, they will be illustrated with an example which is taken from [Sow97a].
This example is a proof of the following syllogism of type Darii:
Every trailer truck has 18 wheels. Some Peterbilt is a trailer truck. Therefore,
some Peterbilt has 18 wheels.
We start with an EG which encodes our premises:
Peterbilt
trailer truck

trailer truck
has 18 wheels

We use the rule of iteration to extend the outer line of identity into the cut:
Peterbilt
trailer truck

trailer truck

has 18 wheels


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2 Short Introduction to Existential Graphs

As the area of this cut is oddly enclosed, the insertion-rule allows us to join
the loose end of the line of identity we have just iterated with the other line
of identity:
Peterbilt
trailer truck

trailer truck
has 18 wheels

Now we can remove the inner instance of ‘is a trailer truck’ with the
deiteration-rule:
Peterbilt

trailer truck
has 18 wheels

Next we are allowed to remove the double cut (the space between the inner
and the outer cut is not empty, but what is written on this area is a ligature
which entirely passes through it, thus the application of the double-cut-rule
is still possible):
Peterbilt

trailer truck

has 18 wheels

Finally we erase the remaining instance of ‘is a trailer truck’ with the erasurerule and obtain:
Peterbilt

has 18 wheels

This is a graph with the meaning ‘some Peterbilt has 18 wheels’, which is the
conclusion of the syllogism.

2.3 Gamma
Gamma shall not be described in detail here. Instead, only some peculiar aspects of Gamma are picked out. The Gamma system was never completed
(in 4.576, Peirce comments Gamma as follows: I was as yet able to gain mere
glimpses, sufficient only to show me its reality, and to rouse my intense curiosity, without giving me any real insight into it.), and it is difficult to be
sure about Peirce’s intention. Roughly speaking, it encompasses higher order
and modal logic, and the possibilty to express self-reference. The probably


2.3 Gamma

15

best-known new device of Gamma is the so-called broken cut. In Fig. 2.1,
two graphs of 4.516 are depicted (the letter ‘g‘ is used by Peirce to denote an
arbitrary graph):

g

g

Fig. 2.1. Figs. 182 and 186 in 4.516

Peirce describes these graphs as follows: Of a certain graph g let us suppose
that I am in such a state of information that it may be true and may be false;
that is I can scribe on the sheet of assertion Figs. 182 and 186. We see that
encircling a graph E by a broken cuts is interpreted as ‘it is possibly not the
case that E holds’, thus, the broken cut corresponds to the syntactical device
‘✸¬’ of modal logic.
Another important aspect of Gamma is the possibility to express meta-level
propositions, i.e. propositions about propositions. As Peirce says: a main idea
of Gamma is that a graph is applicable instead of merely applying it (quotation from [Rob73]). In other words: graphs, which have been used to speak
about objects so far, can now in Gamma be treated like objects themselves
such that other graphs speak about them (this is a kind of abstraction which
Peirce called ‘hypostatic abstraction’, and in modern computer science, often the term ‘reification’ is used). A simple example for this idea can be
found in [Pei92], where Peirce says: When we wish to assert something about
a proposition without asserting the proposition itself, we will enclose it in a
lightly drawn oval, which is supposed to fence it off from the field of assertions.
Peirce provides the following graph to illustrate his explanation:
You are a good girl

is much to be wished

The meaning of this graph is: ‘You are a good girl’ is much to be wished.
Peirce generalized the notation of a cut. The lightly drawn oval is not used
to negate the enclosed graph, it is merely used to fence it off from the field of
assertions and to provide a graphical possibility to speak about it.
Peirce extended this approach further. He started to use colors or tinctures to
distinguish different kind of contexts. Peirce said himself: The nature of the
universe or universes of discourse (for several may be referred to in a single

assertion) in the rather unusual cases in which such precision is required,
is denoted either by using modifications of the heraldic tinctures, marked in
something like the usual manner in pale ink upon the surface, or by scribing
the graphs in colored inks. (quotation taken from [Sow]). For example, in the